Wk1 Notes PDF Lecture Notes

Wittis artifact algebra study of structures the exploration ofimaginativelogic groups operationsstructuresringssets useful forcryptography iii S...

Wittis artifact algebra study of structures the exploration ofimaginativelogic groups operationsstructuresringssets useful forcryptography iii Set Asethas elements sometimes land members ix e s is in a setlands is anelementof s keyfact Anelement iseitherin s ornotin s butnotboth aLawofnoncontradiction excludedminus Example pe IN p is prime p inthe naturalnumbers such that p isprime this is a set Example R theset allsetsthat donot contain themselves of S I S s isthisaset No sentreferential behaviorlike this Russel'sparadox Somecommonsets 1IN natural numbers 0,11213 Eight sina.am 3 rationalnumbers I a be bto ration 4 IR realtumier in comfreisintert atbi I a.be R Definition 1 if A B aresets then Ais a subset of itandonly if every element of A is in B Definition 2 If Aand Baresets Ais equalto B oftheelementsof AandB are equal e'quit Lemma1 Let Aand B be sets A is is itandonlyif A is a subsetg B A B and B is a subset of A cB a have toprove it bothways citandonlyit Lemma2 Leta Bic besets If A B and B c then a c Proof Let a A Because a B weknow a B sinceBecforallbeB wehave bec since a c B a e e andby desinition A c Definition 3 let Aand B besets intheunionof aandB isthethings ineither A or B Aub x xc a or e B  a the intersection of aandB isthesetofthings in aandB ans I aBy 0i Definition 4 It a Baresetsthenthecomplement of a B is theset ofelements in notin a Ac A BIA be B b A 1B 5 Let Aand B beset TheCartesian productof Aand B isthesetof tuples can a ca.be B axB I can s Definition where operations ecture 2 911g Definitionof operations Let A be aset A binaryoperationon a is a race forassigning toeachelementexineaxa exacting aelement yea example I y x xyty binaryoperation IR og incx.gs netbinaryoperation oc.si incon iru a any mincx.gs binaryoperation isbiggest abiciaer 5 ait binaryoperation Given a binaryoperationwecanmakeanoperationtable on o i o o o com cos i o 1 Importantdefinitions regarding operations set A bea settogetherwitha binaryoperation is commutativemeans an baa for anic e a is associativemean cab c a co c for anic e a has an identitymean there exists ee a suchthat a e a and e a a a ca Definition It 1A is a set with binary operation and identity ee A then A has inverses it for all a bicea same ii iii Iii c a Donna Example List Binary Op Commutative Associative Identity 1 R.rs IL 2 Peopueheight shortest 3 I o 4 I 2 313 2 doesn't worksince 01 1 i s 0,13 I I xtxyty 2 0 0 2 7 Ro ofmining a a 2 2matrices